<!-- build time:Sun Nov 24 2019 22:25:13 GMT+0800 (China Standard Time) --><!DOCTYPE html><html lang="zh"><head><meta charset="utf-8"><title>机器学习之线性回归模型 - Note?Note!</title><meta name="viewport" content="width=device-width,initial-scale=1,maximum-scale=1"><meta name="description" content="高斯分布高斯分布的概率密度函数$$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^ {-\frac{(x-\mu)^2}{2 \sigma^2}}\tag{1}$$"><meta name="keywords" content="梯度下降,Sigmoid,Gradient descent,线性回归,高斯分布,最大似然估计"><meta property="og:type" content="article"><meta property="og:title" content="机器学习之线性回归模型"><meta property="og:url" content="http:&#x2F;&#x2F;www.borgor.cn&#x2F;2019-09-25&#x2F;61e41abb.html"><meta property="og:site_name" content="Note?Note!"><meta property="og:description" content="高斯分布高斯分布的概率密度函数$$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^ {-\frac{(x-\mu)^2}{2 \sigma^2}}\tag{1}$$"><meta property="og:locale" content="zh-CN"><meta property="og:image" content="https:&#x2F;&#x2F;imgs.borgor.cn&#x2F;imgs&#x2F;20190926204917.png"><meta 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src="https://cdn.pydata.org/bokeh/release/bokeh-1.3.4.min.js"></script><script type="text/javascript" src="https://cdn.pydata.org/bokeh/release/bokeh-widgets-1.3.4.min.js"></script><style>.bk-root{text-align:center}.bk-root .bk{margin:auto!important}</style><div class="level article-meta is-size-7 is-uppercase is-mobile is-overflow-x-auto"><div class="level-left"><time class="level-item has-text-grey" datetime="2019-09-25T02:58:32.000Z">2019-09-25</time><div class="level-item"><a class="has-link-grey -link" href="/categories/AI/">AI</a>&nbsp;/&nbsp;<a class="has-link-grey -link" href="/categories/AI/Machine-Learning/">Machine Learning</a></div><span class="level-item has-text-grey">34 分钟 读完 (大约 5069 个字) </span><span class="level-item has-text-grey" id="busuanzi_container_page_pv"><i class="far fa-eye"></i> <span id="busuanzi_value_page_pv">0</span>次访问</span></div></div><h1 class="title is-size-3 is-size-4-mobile has-text-weight-normal">机器学习之线性回归模型</h1><div class="content"><h1 id="高斯分布"><a href="#高斯分布" class="headerlink" title="高斯分布"></a>高斯分布</h1><h2 id="高斯分布的概率密度函数"><a href="#高斯分布的概率密度函数" class="headerlink" title="高斯分布的概率密度函数"></a>高斯分布的概率密度函数</h2><p>$$<br>f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^ {-\frac{(x-\mu)^2}{2 \sigma^2}}<br>\tag{1}<br>$$</p><a id="more"></a><p>记做：<br>$$<br>X \sim N(\mu , \sigma ^ 2)<br>\tag{2}<br>$$<br>其函数图像为：</p><div class="bk-root" id="255c804b-c9f6-4cdf-9ef9-e3ba226437b6" data-root-id="1043"></div><script type="application/json" 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Application","version":"1.3.4"}}</script><script type="text/javascript">!function(){var e=function(){Bokeh.safely(function(){!function(e){function n(e){var n=document.getElementById("1125").textContent,o=[{docid:"398620e6-108f-4f46-89fb-af52bfa27cbd",roots:{1043:"255c804b-c9f6-4cdf-9ef9-e3ba226437b6"}}];e.Bokeh.embed.embed_items(n,o)}if(void 0!==e.Bokeh)n(e);else var o=0,t=setInterval(function(e){void 0!==e.Bokeh&&(n(e),clearInterval(t)),o++,o>100&&(console.log("Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing"),clearInterval(t))},10,e)}(window)})};"loading"!=document.readyState?e():document.addEventListener("DOMContentLoaded",e)}()</script>可以看出，正态分布的数学期望值或期望值$\mu$等于位置参数，决定了分布的位置；其方差$\sigma^2$的开平方或这说标准差$\sigma$（标准差的平方等于方差）等于尺度参数，决定了分布的幅度。<h2 id="以上图像的Python代码"><a href="#以上图像的Python代码" class="headerlink" title="以上图像的Python代码"></a>以上图像的Python代码</h2><figure class="highlight python hljs"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br></pre></td><td class="code"><pre><span class="line"><span class="hljs-keyword">import</span> numpy <span class="hljs-keyword">as</span> np</span><br><span class="line"><span class="hljs-keyword">from</span> bokeh.layouts <span class="hljs-keyword">import</span> row, widgetbox</span><br><span class="line"><span class="hljs-keyword">from</span> bokeh.models <span class="hljs-keyword">import</span> CustomJS, Slider</span><br><span class="line"><span class="hljs-keyword">from</span> bokeh.plotting <span class="hljs-keyword">import</span> figure, output_file, show, ColumnDataSource</span><br><span class="line"></span><br><span class="line">N = <span class="hljs-number">100</span></span><br><span class="line">x = np.linspace(<span class="hljs-number">-5.0</span>, <span class="hljs-number">5.0</span>, N)</span><br><span class="line">mu = <span class="hljs-number">0</span></span><br><span class="line">sigma = <span class="hljs-number">3</span></span><br><span class="line">s = (<span class="hljs-number">1</span> / (sigma * np.sqrt(<span class="hljs-number">2</span> * np.pi))) \</span><br><span class="line">    * np.exp(- (np.power(x - mu, <span class="hljs-number">2</span>) / (<span class="hljs-number">2</span> * sigma ** <span class="hljs-number">2</span>)))</span><br><span class="line">source = ColumnDataSource(data=dict(x=x, y=s))</span><br><span class="line">plot = figure(y_range=(<span class="hljs-number">-0.1</span>, <span class="hljs-number">0.4</span>), x_range=(<span class="hljs-number">-5</span>, <span class="hljs-number">5</span>), plot_width=<span class="hljs-number">400</span>, plot_height=<span class="hljs-number">400</span>)</span><br><span class="line">plot.line(<span class="hljs-string">'x'</span>, <span class="hljs-string">'y'</span>, source=source, line_width=<span class="hljs-number">3</span>, line_alpha=<span class="hljs-number">0.6</span>)</span><br><span class="line">callback = CustomJS(args=dict(source=source),</span><br><span class="line">                    code=<span class="hljs-string">"var data = source.data;"</span></span><br><span class="line">                         <span class="hljs-string">"var mu = mu.value;"</span></span><br><span class="line">                         <span class="hljs-string">"var sigma = sigma.value;"</span></span><br><span class="line">                         <span class="hljs-string">"x = data['x'];"</span></span><br><span class="line">                         <span class="hljs-string">"y = data['y'];"</span></span><br><span class="line">                         <span class="hljs-string">"console.log(123);"</span></span><br><span class="line">                         <span class="hljs-string">"for (i = 0; i &lt; x.length; i++) &#123;"</span></span><br><span class="line">                         <span class="hljs-string">"y[i] = (1 / (sigma * Math.sqrt(2 * Math.PI))) * Math.exp(- "</span></span><br><span class="line">                         <span class="hljs-string">"(Math.pow(x[i] - mu, 2) / (2 * sigma * sigma)));"</span></span><br><span class="line">                         <span class="hljs-string">"&#125;"</span></span><br><span class="line">                         <span class="hljs-string">"source.change.emit();"</span>)</span><br><span class="line">sigma_slider = Slider(start=<span class="hljs-number">0.1</span>, end=<span class="hljs-number">10</span>, value=<span class="hljs-number">3</span>, step=<span class="hljs-number">.1</span>,</span><br><span class="line">                      title=<span class="hljs-string">"标准差(σ)"</span>, callback=callback)</span><br><span class="line">callback.args[<span class="hljs-string">"sigma"</span>] = sigma_slider</span><br><span class="line"></span><br><span class="line">mu_slider = Slider(start=<span class="hljs-number">-3</span>, end=<span class="hljs-number">3</span>, value=<span class="hljs-number">0</span>, step=<span class="hljs-number">.1</span>,</span><br><span class="line">                   title=<span class="hljs-string">"期望(μ)"</span>, callback=callback)</span><br><span class="line">callback.args[<span class="hljs-string">"mu"</span>] = mu_slider</span><br><span class="line">layout = row(</span><br><span class="line">    plot,</span><br><span class="line">    widgetbox(sigma_slider, mu_slider),</span><br><span class="line">)</span><br><span class="line">output_file(<span class="hljs-string">"../output/html/04/slider.html"</span>, title=<span class="hljs-string">"正态分布概率密度函数"</span>)</span><br><span class="line">show(layout)</span><br></pre></td></tr></table></figure><h2 id="二维高斯分布概率密度函数"><a href="#二维高斯分布概率密度函数" class="headerlink" title="二维高斯分布概率密度函数"></a>二维高斯分布概率密度函数</h2><p>$$<br>f(x,y) = \left ( 2 \pi \sigma_1\sigma_2 \sqrt{1-\rho^2} \right)^{-1}<br>\exp{\left \lbrace - \frac{1}{2(1-\rho^2)} \left [ \frac{(x - \mu_1)^2}{\sigma_1^2} - \frac{2\rho(x-\mu_1)(x-\mu_2)}{\sigma_1 \sigma_2} + \frac{(x - \mu_2)^2}{\sigma_2^2} \right]<br>\right \rbrace }<br>\tag{3}<br>$$</p><p>其中$\mu_1,\mu_2,\sigma_1,\sigma_2,\rho$都是常数，$x$是自变量，我们称作$X_1,X_2$服从参数为$\mu_1,\mu_2,\sigma_1,\sigma_2,\rho$的二维正态分布，常把这个分布记作：<br>$$<br>(X_1,X_2) \sim N(\mu_1,\mu_2,\sigma_1^2,\sigma_2^2,\rho)\tag{4}<br>$$<br>的范围分别为：<br>$$<br>\begin{align}<br>\\\ - \infty \lt &amp; \mu_1 \lt + \infty<br>\\\ - \infty \lt &amp; \mu_2 \lt + \infty<br>\\\ -1 \lt &amp; \rho \lt 1<br>\\\ \sigma_1 &amp; \ge 0<br>\\\ \sigma_2 &amp; \ge 0<br>\end{align} \tag{4.1}<br>$$<br>其函数图像为：</p><p><img src="https://imgs.borgor.cn/imgs/20190925214942.png" alt="标准二维正态分布图像"></p><h1 id="最大似然估计-MLE"><a href="#最大似然估计-MLE" class="headerlink" title="最大似然估计(MLE)"></a>最大似然估计(MLE)</h1><p><img src="https://imgs.borgor.cn/imgs/20190925215322.png" alt="https://blog.csdn.net/zengxiantao1994/article/details/72787849"></p><blockquote><p>注：最大似然估计在我看来就是，将在先验经验中出现的最多的分类结果作为分类依据。</p><p>比如说，有十张照片，都是狗狗，这个是先验经验，我们从中抽取出一种特征：两个眼睛，与之关联起来之后，我们可以说：两个眼睛~狗</p><p>但是在继续训练的时候，我们发现，猫也有两个眼睛，这会造成分类错误，所以添加更多的特征，比如：会汪汪叫。结果，发现这个正确概率率提高了不少，除了少数误差数据，比如鹦鹉学舌、人声之类的。我们几乎可以肯定（两个眼睛, 汪汪叫）~ 狗。</p><p>注：正确理解正确率的含义，就是正确的概率。</p></blockquote><p>原理：极大似然估计是建立在极大似然原理的基础上的一个统计方法，是概率论在统计学中的应用。极大似然估计提供了一种给定观察数据来评估模型参数的方法，即：“模型已定，参数未知”。通过若干次试验，观察其结果，利用试验结果得到某个参数值能够使样本出现的概率为最大，则称为极大似然估计。</p><h2 id="数学表述"><a href="#数学表述" class="headerlink" title="数学表述"></a>数学表述</h2><p>由于样本集中的样本都是独立同分布，可以只考虑一类样本集$D$，来估计参数向量$\theta$。记已知的样本集为：<br>$$<br>D = \lbrace x_1, x_2, \dots, x_N \rbrace<br>\tag{5}<br>$$<br>似然函数（$linkehood \quad function$）：联合概率密度函数$p(D|\theta)$称为相对于$\lbrace x_1, x_2, \dots, x_N \rbrace$的$\theta$的似然函数。<br>$$<br>l(\theta) = p(D| \theta) = p(x_1,x_2,\dots,x_N) = \prod_{i=1}^N p(D|\theta)<br>\tag{5}<br>$$<br>如果$\hat{\theta}$是参数空间中能使似然函数$l(\theta)$最大的θ值，则$\hat{\theta}$应该是<strong>“最可能”</strong>的参数值，那么$\hat{\theta}$就是$\theta$的极大似然估计量。它是样本集的函数，记作：<br>$$<br>\hat{\theta} = d(x_1,x_2,\dots, x_N)=d(D)<br>\tag{6}<br>$$<br>其中，$\theta(x_1,x_2,\dots, x_N)$称作极（最）大似然估计函数的估计值。</p><h1 id="最小二乘法"><a href="#最小二乘法" class="headerlink" title="最小二乘法"></a>最小二乘法</h1><blockquote><p>以下来自维基百科</p></blockquote><p>某次实验得到了四个数据点:$ (1,6),(2,5),(3,7),(4,10)$。我们希望找出一条和这四个点最匹配的直线$y=\theta_1 + \theta_2 x$，即找出在某种“最佳情况”下能够大致符合如下超定线性方程组的$\beta_1$和$\beta_2$：</p><p>应为有四组数据，我们可以将方程组表达为：<br>$$<br>y = \theta_0x_0 + \theta_1 x _1 \\\<br>\tag{7.1}<br>$$</p><p>$$<br>y = \theta x<br>\tag{7.2}<br>$$</p><p>$7.2$是$7.1$的向量表示方法。</p><p>在例子中，我们的方程组为：<br>$$<br>\begin{align}<br>x_1 + x_2 &amp; = 6 \\\<br>x_1 + 2x_2 &amp; = 5 \\\<br>x_1 + 3x_2 &amp; = 6 \\\<br>x_1 + 4x_2 &amp; = 10 \\\<br>\end{align}<br>\tag{8}<br>$$<br>表达为向量：<br>$$<br>\left [<br>\begin{matrix}<br>1 &amp; 1 \\\<br>1 &amp; 2 \\\<br>1 &amp; 3 \\\<br>1 &amp; 4<br>\end{matrix}<br>\right] \left [<br>\begin{matrix}<br>x_1 \\\<br>x_2<br>\end{matrix}<br>\right ] = \left[<br>\begin{matrix}<br>6 \\\<br>5 \\\<br>7 \\\<br>10<br>\end{matrix}<br>\right]<br>\tag{9}<br>$$<br>最小二乘法采用的手段是尽量使得等号两边的方差最小，也就是找出下面这个函数的最小值：<br>$$<br>S(\beta_1, \beta_2) = [6-(\beta_1 + \beta_2)]^2 + [5-(\beta_1 + 2\beta_2)]^2 +<br>[7-(\beta_1 + 3\beta_2)]^2 + [10-(\beta_1 + 4\beta_2)]^2 \tag{10}<br>$$</p><blockquote><p>对于我们的拟合曲线来说，上面的方差函数就是我们所说的损失函数的一种。其中的每一项，如$6-(\beta_1+\beta_2)$就是其偏差。为防止正负偏差互相抵消，所以对其进行平方操作，然后求所有偏差之和的最小值，就是拟合最好的情况，</p></blockquote><p>对两边分别求两个参数的偏导数：<br>$$<br>\frac{\partial S }{\partial\beta_1} = 0 = 8\beta_1 + 20 \beta_2 + 56 \\\<br>\frac{\partial S}{\partial \beta_2} = 0 = 20\beta_1 + 60\beta_2 + 154<br>\tag{11}<br>$$<br>求解方程组，得到：<br>$$<br>\beta_1 = 3.5 \\\<br>\beta_2 = 1.4<br>\tag{12}<br>$$<br>如此就得到了一个只有两个未知数的方程组，很容易就可以解出：根据结论。最小值可以通过对$S(\beta_1, \beta_2) $分别求$\beta_1$和$\beta_2$的偏导数偏导数)，然后使它们等于零得到。</p><p><img src="https://imgs.borgor.cn/imgs/20190926083344.png" alt="图片来自维基百科"></p><p>也就是说直线$y=3.5 + 1.4 x$ 是最佳的。</p><h1 id="Logistic-回归"><a href="#Logistic-回归" class="headerlink" title="$Logistic$回归"></a>$Logistic$回归</h1><h2 id="Sigmoid-函数"><a href="#Sigmoid-函数" class="headerlink" title="$Sigmoid$函数"></a>$Sigmoid$函数</h2><div class="bk-root" id="65c3b379-fc29-489d-aa67-abbad1fcf97f" data-root-id="1002"></div><script type="application/json" 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Application","version":"1.3.4"}}</script><script type="text/javascript">!function(){var e=function(){Bokeh.safely(function(){!function(e){function n(e){var n=document.getElementById("1128").textContent,o=[{docid:"7344412d-7bc9-4a59-8502-cdd13ad92044",roots:{1002:"65c3b379-fc29-489d-aa67-abbad1fcf97f"}}];e.Bokeh.embed.embed_items(n,o)}if(void 0!==e.Bokeh)n(e);else var o=0,t=setInterval(function(e){void 0!==e.Bokeh&&(n(e),clearInterval(t)),o++,o>100&&(console.log("Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing"),clearInterval(t))},10,e)}(window)})};"loading"!=document.readyState?e():document.addEventListener("DOMContentLoaded",e)}()</script>$Sigmoid$函数其实是将整个实数轴上的所有实数全部映射到$0 \sim 1$之间，即：$[- \infty, + \infty] \sim [0,1]$，在绝对值很大的时候，会将其映射的值贴近与0或者1。<h2 id="Logestic-回归"><a href="#Logestic-回归" class="headerlink" title="$Logestic$ 回归"></a>$Logestic$ 回归</h2><p>$Logestic$回归的假设函数如下：<br>$$<br>\begin{align}<br>h_\theta（x）&amp; = g(\theta^T x) \\\<br>g(z) &amp; = \frac{1}{1+e^{-z}}<br>\end{align}<br>\tag{13}<br>$$<br>将其合并为一个公式：<br>$$<br>h_\theta = \frac{1}{1+e^{-\theta ^T x}}<br>\tag{14}<br>$$<br>其中$x$是我们的输入，$\theta$为我们要求取的参数。</p><p>一个机器学习的模型，实际上是把决策函数限定在某一组条件下，这组限定条件就决定了模型的假设空间。当然，我们还希望这组限定条件简单而合理。而逻辑回归模型所做的假设是：<br>$$<br>P( y = 1 | x;\theta) = g(\theta ^Tx) = \frac{1}{1+e^{-\theta ^Tx}}<br>\tag{15}<br>$$<br>这个函数的意思就是在给定$x$和$\theta$的条件下$y=1$的概率。</p><p>这里$g(h)$就是我们上面提到的$sigmoid$函数，与之相对应的决策函数为：<br>$$<br>y^* = 1, ifP(y=1|x) \gt 0.5<br>\tag{16}<br>$$<br>选择$0.5$作为阈值是一个一般的做法，实际应用时特定的情况可以选择不同阈值，如果对正例的判别准确性要求高，可以选择阈值大一些，对正例的召回要求高，则可以选择阈值小一些。</p><h1 id="梯度下降算法"><a href="#梯度下降算法" class="headerlink" title="梯度下降算法"></a>梯度下降算法</h1><p>$$<br>\Theta^1 = \Theta^0 - \alpha \nabla J (\Theta) \\\<br>evaluated \quad at \quad \Theta^0<br>\tag{17}<br>$$</p><p>此公式的意义是：$J$是关于$\Theta$的一个函数，我们当前所处的位置为$\Theta^0$点，要从这个点走到$J$的最小值点(此处有可能是极小值二不是最小值)。首先我们先确定前进的方向，也就是梯度的反向，然后走一段距离的步长，也就是$\alpha$，走完这个段步长，就到达了$\Theta^1$这个点！</p><h1 id="损失函数的选择"><a href="#损失函数的选择" class="headerlink" title="损失函数的选择"></a>损失函数的选择</h1><p>对于训练样本来说，我们选择了一条曲线：<br>$$<br>\hat{y}_i = \theta_0 + \theta_1 x<br>\tag{18}<br>$$<br>作为其拟合曲线。</p><p>我们为其构造了一个损失函数：<br>$$<br>C = \sum_{i=1}^n (y_i - \hat{y}_i)^2<br>\tag{19}<br>$$<br>表示每个训练数据点$（x_i, y_i）$到拟合直线$\hat{y_i} = \theta_0 + \theta_1 x$的竖直距离的平方和，通过最小化这个损失函数来求得拟合直线的最佳参数$\mathbb{\theta}$，实际上就是求损失函数$C$在取得最小值情况下$\mathbb{\theta}$的值。<strong>那么损失函数为什么要用平方差形式呢</strong>，而不是绝对值形式，一次方，三次方，或四次方形式？</p><p>简单的说，是因为使用平方形式的时候，使用的是<strong>“最小二乘法”</strong>的思想，这里的“二乘”指的是用平方来度量观测点与估计点的距离（远近），“最小”指的是参数值要保证各个观测点与估计点的距离的平方和达到最小。</p><p>我们设观测输出与预估数据之间的误差为：</p><p>$$<br>{\varepsilon _i} = {y_i} - {\widehat y_i}<br>\tag{20}<br>$$</p><p>我们通常认为 $\varepsilon$ 服从正态分布，即：<br>$$<br>f(\varepsilon _i;u,\sigma ^2) = \frac{1}{\sigma \sqrt {2\pi } }\bullet \exp \left [ - \frac{(\varepsilon _i - u)^2}{2{\sigma ^2}} \right ]<br>\tag{21}<br>$$</p><p>我们求的参数$\varepsilon$的极大似然估计$(\mu, \sigma^2)$，即是说，在某个$(\mu, \sigma^2)$下，使得服从正态分布的$\varepsilon$取得现有样本$\varepsilon$的概率最大。那么根据极大似然估计函数的定义，令：</p><p>$$<br>L(\mu,\sigma^2)=\prod_{i=1}^n \frac{1}{\sqrt{2 \pi}\sigma} \bullet \exp{(-\frac{(\varepsilon_i - \mu)^2}{2 \sigma^2})}<br>\tag{22}<br>$$</p><p>取对数似然函数：<br>$$<br>\log L(\mu, \sigma^2) = -\frac{n}{2} \log \sigma^2 - \frac{n}{2} \log 2 \pi - \frac{\sum_{i=1}^n (\varepsilon_i - \mu)^2}{2 \sigma^2}<br>\tag{23}<br>$$</p><p>分别求$(\mu, \sigma^2)$的偏导数，然后置$0$，最后求得参数$(\mu, \sigma^2)$的极大似然估计为：</p><p>$$<br>\mu = \frac{1}{n} \sum_{i=1}{n} \varepsilon_i<br>\tag{24}<br>$$</p><p>$$<br>\sigma ^ 2 = \frac{1}{n} \sum_{i=1}^n (\varepsilon_i - \mu) ^ 2<br>\tag{25}<br>$$</p><p>我们在线性回归中要求得最佳拟合直线$\hat{y_i} = \theta_0 + \theta _ 1 x$，实质上是求预估值$\hat{y_i}$与观测值$y_i$之间的误差$\varepsilon$最小（最好是没有误差）的情况下$\theta$的值。而前面提到过，$\varepsilon$是服从参数$(\mu, \sigma^2)$的正态分布，那最好是均值$\mu$和方差$\sigma$趋近于$0$或越小越好。即:</p><p>$$<br>\mu = \frac{1}{n} \sum_{i=1}^n \varepsilon_i = \frac{1}{n} \sum_{i=1}^n (y_i - \hat{y}_i) \to 0 (越小越好)<br>\tag{26})<br>$$</p><p>$$<br>\sigma ^2 = \frac{1}{n} \sum_{i = 1}^n (\varepsilon_i - \mu)^2 = \frac{1}{n} \sum_{i=1}^n (y_i - \widehat y_i - \mu)^2 \approx \frac{1}{n} \sum _{i = 1}^n (y_i - \widehat{y_i})^2<br>\tag{27}<br>$$<br>而这与最前面构建的平方形式损失函数本质上是等价的。</p><script type="text/javascript" src="https://cdn.jsdelivr.net/npm/kity@2.0.4/dist/kity.min.js"></script><script type="text/javascript" src="https://cdn.jsdelivr.net/npm/kityminder-core@1.4.50/dist/kityminder.core.min.js"></script><script defer type="text/javascript" src="https://cdn.jsdelivr.net/npm/hexo-simple-mindmap@0.2.0/dist/mindmap.min.js"></script><link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/npm/hexo-simple-mindmap@0.2.0/dist/mindmap.min.css"></div><div class="level is-size-7 is-uppercase"><div class="level-start"><div class="level-item"><span class="is-size-6 has-text-grey has-mr-7">#</span> <a class="has-link-grey -link" href="/tags/Gradient-descent/" rel="tag">Gradient descent</a>, <a class="has-link-grey -link" href="/tags/Sigmoid/" rel="tag">Sigmoid</a>, <a class="has-link-grey -link" href="/tags/%E6%9C%80%E5%A4%A7%E4%BC%BC%E7%84%B6%E4%BC%B0%E8%AE%A1/" rel="tag">最大似然估计</a>, <a class="has-link-grey -link" href="/tags/%E6%A2%AF%E5%BA%A6%E4%B8%8B%E9%99%8D/" rel="tag">梯度下降</a>, <a class="has-link-grey -link" href="/tags/%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/" rel="tag">线性回归</a>, <a class="has-link-grey -link" href="/tags/%E9%AB%98%E6%96%AF%E5%88%86%E5%B8%83/" rel="tag">高斯分布</a></div></div></div></div></div><div class="card"><div class="card-content"><h3 class="menu-label has-text-centered">喜欢这篇文章？打赏一下作者吧</h3><div class="buttons is-centered"><a class="button is-info donate"><span class="icon is-small"><i class="fab fa-alipay"></i> </span><span>支付宝</span><div class="qrcode"><img src="https://imgs.borgor.cn/imgs20190628231540.png" alt="支付宝"></div></a><a class="button is-success donate"><span class="icon is-small"><i class="fab fa-weixin"></i> </span><span>微信</span><div 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class="level-end"><span class="level-item tag">9</span></span></a></li><li><a class="level is-marginless" href="/categories/Development/React/"><span class="level-start"><span class="level-item">React</span> </span><span class="level-end"><span class="level-item tag">2</span></span></a></li><li><a class="level is-marginless" href="/categories/Development/html5/"><span class="level-start"><span class="level-item">html5</span> </span><span class="level-end"><span class="level-item tag">1</span></span></a></li></ul></li><li><a class="level is-marginless" href="/categories/Operations/"><span class="level-start"><span class="level-item">Operations</span> </span><span class="level-end"><span class="level-item tag">18</span></span></a><ul><li><a class="level is-marginless" href="/categories/Operations/Nginx/"><span class="level-start"><span class="level-item">Nginx</span> </span><span class="level-end"><span class="level-item tag">1</span></span></a></li><li><a class="level is-marginless" 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